The Kelly criterion, as a bet-sizing optimum, makes a few assumptions, which are not true in most humans.
Future bets will be available, but limited by the results of the currently-considered wager. That is, there is a bankroll which can grow and shrink, but if it hits zero, the model ends. Kelly phrases this requirement as “the possibility of reinvestment”.
Utility of money is logarithmic in the ranges under consideration (that is, you’re considering lifetime resources, not just the amount in your pocket right now).
The gambler introduced here follows an essentially different criterion from the classical gambler. At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with 926 the bell system technical journal, july 1956 the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies. Suppose the situation were different; for example, suppose the gambler’s wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet. He would bet all his available capital (one dollar) on the event yielding the highest expectation. With probability one he would get ahead of anyone dividing his money differently.
This all implies that the special-case is the last wager you will ever make. And from there the more complicated cases of the penultimate wager and the probabilistic-finite cases. I don’t know how big the chain needs to get to converge to Kelly being the optimum, but since it’s compatible with logarithmic utility of money in the first place, for some agents it’ll be the same regardless.
My strong suspicion is that Kelly always applies if your terminal utility function for money is logarithmic. But I don’t see how that could be—the marginal amount of money/resources you’ll control at death is tiny compared to all resources in the universe, so your utility for any margin under consideration should be close to linear.
My strong suspicion is that Kelly always applies if your terminal utility function for money is logarithmic. But I don’t see how that could be—the marginal amount of money/resources you’ll control at death is tiny compared to all resources in the universe, so your utility for any margin under consideration should be close to linear.
If the amount is tiny, and your utility is log resources, then that puts us close to the origin, where the derivative of the logarithm is very high, and reducing very quickly.
But logarithm can still look nearly linear if the differences we can make are sufficiently small in relation to the total.
Sure, but the point of the Kelly calculation is to PICK the amount, relative to the potential gain and risk of ruin. Which ends up equivalent to logarithmic utility.
For the final bet (or the induction base for a finite sequence), one cannot pick an amount without knowing the zero-point on the utility curve.
What my comment about small differences amounts to is if you can’t bet the whole bankroll; for example if your utility is (altruistically) logarithmic in the resources of all humanity, but you can only control how to gamble with a small fraction of that.
This might justify EAs behaving like their utility is linear in resources, even if it’s ultimately logarithmic.
The problem is in the “I have” statement in the setup. After your final bet, you will be dead (or at least not have any ability for further decisions about the money). You have to specify what “have” means, in terms of your utility. Perhaps you’ve willed it all to a home for cats, in that case the home has 500,100 +/- x. Perhaps you care about humanity as a whole, in which case your wager has no impact—any that you add or remove from “your” $100 comes out of someone else’s. Or if the wager is making something worth x, or destroying x value as your final act, then humanity as a whole has $90T +/- x.
As far as I can tell, the fact that you only ever control a very small proportion of the total wealth in the universe isn’t something we need to consider here.
No matter what your wealth is, someone with log utility will treat a prospect of doubling their money to be exactly as good as it would be bad to have their wealth cut in half, right?
I don’t know—I don’t have a good sense of what “terminal values” mean for humans. But I suspect it does matter—for a logarithmic utility curve, figuring out the change in utility for a given delta in resources depends entirely on the proportion of the total that the given delta represents.
makes a few assumptions, which are not true in most humans
Right, it would be interesting to take a Kelly-like approach while relaxing those assumptions.
Fixed income: most people have money coming in through work, not just investment. This should make a person less risk-averse, intuitively, since losing all one’s money no longer means you’re out of the game forever.
Fixed expenses: on the flip side, most people have expenses which are relatively fixed in the short term (and increase with bankroll in the longer term, reflecting a desire to spend money to get other things!)
Fixed income: most people have money coming in through work, not just investment.
This is a very common situation, and the standard recommendation for Kelly criterion usage is “calculate based on your ENTIRE bankroll”. Yes, the point on the logarithm is based on your home equity and expected future earnings, not just the money in your pocket.
This usually translates to “if the expectation is positive, bet the maximum”. Most people don’t think that way, and therefore don’t optimize their lifetime bankroll growth. Also, cases where you actually know that it’s +EV with enough certainty to use Kelly are extremely rare.
The Kelly criterion, as a bet-sizing optimum, makes a few assumptions, which are not true in most humans.
Future bets will be available, but limited by the results of the currently-considered wager. That is, there is a bankroll which can grow and shrink, but if it hits zero, the model ends. Kelly phrases this requirement as “the possibility of reinvestment”.
Utility of money is logarithmic in the ranges under consideration (that is, you’re considering lifetime resources, not just the amount in your pocket right now).
It’s a little unclear whether the log utility is an assumption or a result of the bankrupcy-is-death assumption. The original paper, http://www.herrold.com/brokerage/kelly.pdf , says:
This all implies that the special-case is the last wager you will ever make. And from there the more complicated cases of the penultimate wager and the probabilistic-finite cases. I don’t know how big the chain needs to get to converge to Kelly being the optimum, but since it’s compatible with logarithmic utility of money in the first place, for some agents it’ll be the same regardless.
My strong suspicion is that Kelly always applies if your terminal utility function for money is logarithmic. But I don’t see how that could be—the marginal amount of money/resources you’ll control at death is tiny compared to all resources in the universe, so your utility for any margin under consideration should be close to linear.
If the amount is tiny, and your utility is log resources, then that puts us close to the origin, where the derivative of the logarithm is very high, and reducing very quickly.
But logarithm can still look nearly linear if the differences we can make are sufficiently small in relation to the total.
Sure, but the point of the Kelly calculation is to PICK the amount, relative to the potential gain and risk of ruin. Which ends up equivalent to logarithmic utility.
For the final bet (or the induction base for a finite sequence), one cannot pick an amount without knowing the zero-point on the utility curve.
Agreed.
What my comment about small differences amounts to is if you can’t bet the whole bankroll; for example if your utility is (altruistically) logarithmic in the resources of all humanity, but you can only control how to gamble with a small fraction of that.
This might justify EAs behaving like their utility is linear in resources, even if it’s ultimately logarithmic.
I’m a little confused about what you mean sorry -
What’s wrong with this example?:
It’s time for the final bet, I have $100 and my utility is U=ln($)
I have the opportunity to bet on a coin which lands heads with probability 34, at 1:1 odds.
If I bet $x on heads, then my expected utility is E(U)=34ln(100+x)+14ln(100−x), which is maximized when x=50.
So I decide to bet 50 dollars.
What am I missing here?
The problem is in the “I have” statement in the setup. After your final bet, you will be dead (or at least not have any ability for further decisions about the money). You have to specify what “have” means, in terms of your utility. Perhaps you’ve willed it all to a home for cats, in that case the home has 500,100 +/- x. Perhaps you care about humanity as a whole, in which case your wager has no impact—any that you add or remove from “your” $100 comes out of someone else’s. Or if the wager is making something worth x, or destroying x value as your final act, then humanity as a whole has $90T +/- x.
As far as I can tell, the fact that you only ever control a very small proportion of the total wealth in the universe isn’t something we need to consider here.
No matter what your wealth is, someone with log utility will treat a prospect of doubling their money to be exactly as good as it would be bad to have their wealth cut in half, right?
I don’t know—I don’t have a good sense of what “terminal values” mean for humans. But I suspect it does matter—for a logarithmic utility curve, figuring out the change in utility for a given delta in resources depends entirely on the proportion of the total that the given delta represents.
Right, it would be interesting to take a Kelly-like approach while relaxing those assumptions.
Fixed income: most people have money coming in through work, not just investment. This should make a person less risk-averse, intuitively, since losing all one’s money no longer means you’re out of the game forever.
Fixed expenses: on the flip side, most people have expenses which are relatively fixed in the short term (and increase with bankroll in the longer term, reflecting a desire to spend money to get other things!)
This is a very common situation, and the standard recommendation for Kelly criterion usage is “calculate based on your ENTIRE bankroll”. Yes, the point on the logarithm is based on your home equity and expected future earnings, not just the money in your pocket.
This usually translates to “if the expectation is positive, bet the maximum”. Most people don’t think that way, and therefore don’t optimize their lifetime bankroll growth. Also, cases where you actually know that it’s +EV with enough certainty to use Kelly are extremely rare.